Quantum Codes with Transversal CCZ Gates and Sublinear Z-Stabilizers
Pith reviewed 2026-06-26 09:42 UTC · model grok-4.3
The pith
Refined puncturing of algebraic expander codes produces explicit CSS quantum codes with transversal CCZ gates, linear dimension, and sublinear-weight Z-stabilizers.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By replacing the global dual-distance assumption in the Guruswami-Golowich puncturing framework with a condition only on the selected puncturing set, the algebraic expander codes of KT26 can be punctured to yield CSS codes with transversal CCZ gates, parameters [[N, Θ(N), Ω(N^{1/m})]] for fixed m ≥ 3, and Z-stabilizer generating sets of weight O(N^{1/m}). The fixed-field versions retain near-linear dimension and sublinear stabilizer locality up to polylog factors.
What carries the argument
The refined puncturing theorem, which relaxes the dual-distance condition to depend only on the puncturing set while preserving the Schur-product and distance properties needed for transversal CCZ.
If this is right
- Explicit growing-alphabet CSS codes exist with linear dimension, polynomial distance, and transversal CCZ gates.
- Z-stabilizers admit explicit generating sets of weight O(N to the power 1/m).
- Fixed prime field versions achieve near-linear dimension and distance n to the 1/m up to polylog factors with sublinear Z-stabilizer locality.
- Transversal CCZ remains available after alphabet reduction via projective-multiplicity multiplication-friendly codes.
Where Pith is reading between the lines
- These codes may enable fault-tolerant implementations of CCZ with lower overhead than previous constructions if the distance and locality scale well in practice.
- Similar refinements could apply to other classical code families with small dual distance but good local properties.
- Testing the refined theorem on smaller instances of algebraic expanders would verify the puncturing set conditions.
- The polylog losses in the fixed-field case suggest room for further optimization in the alphabet reduction step.
Load-bearing premise
The refined puncturing theorem applies to the algebraic expander codes when the puncturing set is chosen to satisfy the local dual-distance condition.
What would settle it
A counterexample algebraic expander code where no suitable puncturing set meets the Schur-product and distance conditions after refinement, or explicit computation showing the resulting quantum distance falls below Omega(N to the 1/m).
read the original abstract
We construct quantum CSS codes with transversal \(CCZ\) gates whose \(Z\)-stabilizers admit sublinear-weight generating sets. We build on the algebraic puncturing framework of Guruswami and Golowich \cite{GG24}, which turns classical codes with the required Schur-product and distance conditions into CSS codes with transversal \(CCZ\). However, applying the framework directly to the algebraic expander codes of \cite{KT26} runs into their small dual distance, and therefore produces only sublinear quantum dimension. Our main technical step is a refined puncturing theorem in which the global dual-distance assumption is replaced by a condition only on the selected puncturing set. Applying this theorem to algebraic expander codes gives explicit growing-alphabet CSS codes with parameters \([[N,\Theta(N),\Omega(N^{1/m})]]\), for every fixed \(m\geq 3\), and with transversal \(CCZ\) gates. Moreover, the \(Z\)-stabilizer space has an explicit generating set of weight \(O(N^{1/m})\). We also reduce the alphabet to a fixed prime field using a projective-multiplicity version of multiplication-friendly codes. The resulting fixed-prime-field CSS code triples, of length \(n\), still have transversal \(CCZ\) gates. Their dimension is near-linear, their distance is \(n^{1/m}\) up to polylogarithmic factors, and the \(Z\)-stabilizer locality remains sublinear, again up to polylogarithmic losses.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript constructs explicit quantum CSS codes with transversal CCZ gates and sublinear-weight Z-stabilizer generators. Building on the Guruswami-Golowich puncturing framework, it introduces a refined puncturing theorem that replaces the global dual-distance assumption with a local condition on the chosen puncturing set S. Applying this to the algebraic expander codes of KT26 produces growing-alphabet CSS codes with parameters [[N, Θ(N), Ω(N^{1/m})]] for every fixed m ≥ 3; a projective-multiplicity reduction then yields fixed-prime-field versions with near-linear dimension, distance n^{1/m} (up to polylog factors), and sublinear Z-stabilizer locality, all while preserving the transversal CCZ property.
Significance. If the refined theorem holds and the KT26 codes satisfy the local condition while retaining Schur-product closure and distance, the work supplies the first explicit constructions of quantum codes with transversal non-Clifford gates at these parameters. This is significant for fault-tolerant quantum computing, where transversal CCZ enables magic-state distillation without magic-state factories. The sublinear Z-stabilizer weights and the explicit alphabet-reduction step are additional strengths; the paper gives credit to the GG24 framework and KT26 expanders while extending them.
major comments (2)
- [Refined puncturing theorem] Refined puncturing theorem (main technical section): the claim that a local dual-distance condition on S suffices to preserve both the Schur-product property and the Ω(N^{1/m}) distance bound must be accompanied by an explicit verification (or existence argument) that such an S exists for the KT26 family without degrading the parameters below the stated thresholds; this step is load-bearing for the central parameter claim.
- [Application to KT26 codes] Application to KT26 codes (parameter derivation): the manuscript states that the punctured codes retain the required minimum distance and Schur closure, but no concrete bound or calculation is supplied showing how the local condition on S implies the final distance Ω(N^{1/m}) (or the near-linear dimension after alphabet reduction); this is needed to support the strongest claim.
minor comments (1)
- [Abstract] Abstract and introduction: the distinction between the growing-alphabet length N and the fixed-field length n is clear in the text but could be emphasized earlier to avoid reader confusion when comparing the two families.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to strengthen the presentation of the refined puncturing theorem and its application.
read point-by-point responses
-
Referee: [Refined puncturing theorem] Refined puncturing theorem (main technical section): the claim that a local dual-distance condition on S suffices to preserve both the Schur-product property and the Ω(N^{1/m}) distance bound must be accompanied by an explicit verification (or existence argument) that such an S exists for the KT26 family without degrading the parameters below the stated thresholds; this step is load-bearing for the central parameter claim.
Authors: We agree that an explicit verification or existence argument for a suitable puncturing set S in the KT26 family is required to fully support the central claims. In the revised manuscript we will add a dedicated subsection providing a probabilistic existence argument (or explicit construction when possible) that such an S exists, drawing on the algebraic structure and expansion properties of the KT26 codes to ensure the local dual-distance condition holds while preserving Schur-product closure and the Ω(N^{1/m}) distance. revision: yes
-
Referee: [Application to KT26 codes] Application to KT26 codes (parameter derivation): the manuscript states that the punctured codes retain the required minimum distance and Schur closure, but no concrete bound or calculation is supplied showing how the local condition on S implies the final distance Ω(N^{1/m}) (or the near-linear dimension after alphabet reduction); this is needed to support the strongest claim.
Authors: We acknowledge that the current manuscript would be strengthened by explicit calculations connecting the local condition on S to the final parameters. In the revision we will insert a parameter-derivation subsection that applies the refined puncturing theorem step-by-step, showing how the local dual-distance condition on S yields the stated Ω(N^{1/m}) distance for the growing-alphabet codes and, after the projective-multiplicity alphabet reduction, the near-linear dimension together with distance n^{1/m} (up to polylog factors). revision: yes
Circularity Check
No significant circularity; new refined theorem supplies independent content.
full rationale
The derivation rests on a newly stated refined puncturing theorem that replaces the global dual-distance hypothesis of GG24 with a local condition on the chosen puncturing set S. This theorem is proved in the present paper and is not defined in terms of the target quantum parameters. The algebraic expander codes of KT26 are invoked only as an external input family that is asserted to satisfy the new local condition together with the required Schur-product and distance properties; no equation or parameter in the paper is fitted to the output dimension or distance, and no self-citation chain is used to force the claimed [[N, Θ(N), Ω(N^{1/m})]] bounds. The construction is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Algebraic expander codes from KT26 satisfy the Schur-product and distance conditions needed for the puncturing framework after the refined selection.
- ad hoc to paper The refined puncturing theorem correctly replaces the global dual-distance assumption with a condition only on the selected puncturing set.
Reference graph
Works this paper leans on
-
[1]
Physical Review Letters , volume =
Eastin, Bryan and Knill, Emanuel , title =. Physical Review Letters , volume =. 2009 , doi =
2009
-
[2]
Physical Review A , volume =
Bravyi, Sergey and Haah, Jeongwan , title =. Physical Review A , volume =. 2012 , doi =
2012
-
[3]
Physical Review Letters , volume =
Krishna, Anirudh and Tillich, Jean-Pierre , title =. Physical Review Letters , volume =. 2019 , doi =
2019
-
[4]
Optimal Proof Systems for Complex Sets Are Hard to Find , year =
Nguyen, Quynh T. , title =. Proceedings of the 57th Annual ACM Symposium on Theory of Computing , series =. 2025 , publisher =. doi:10.1145/3717823.3718186 , archivePrefix =. 2408.10140 , primaryClass =
-
[5]
Proceedings of the 57th Annual ACM Symposium on Theory of Computing , series =
Golowich, Louis and Lin, Ting-Chun , title =. Proceedings of the 57th Annual ACM Symposium on Theory of Computing , series =. 2025 , publisher =. doi:10.1145/3717823.3718139 , archivePrefix =. 2410.14662 , primaryClass =
-
[6]
Golowich, Louis and Guruswami, Venkatesan , title =. 2025 , archivePrefix =. 2510.06798 , primaryClass =
arXiv 2025
-
[7]
Li, Yiming and Li, Zimu and Liu, Zi-Wen , title =. 2026 , archivePrefix =. 2604.01874 , primaryClass =
arXiv 2026
-
[8]
Kopparty, Swastik and Tamo, Itzhak , title =. 2026 , archivePrefix =. 2603.24788 , primaryClass =
arXiv 2026
-
[9]
and Gottesman, Daniel and Krishna, Anirudh , title =
Cui, Shawn X. and Gottesman, Daniel and Krishna, Anirudh , title =. Physical Review A , volume =. 2017 , month = jan, publisher =
2017
-
[10]
Tanner, R. M. , title =. IEEE Transactions on Information Theory , volume =. 1981 , month = sep, doi =
1981
-
[11]
2025 , eprint =
Zhu, Guanyu , title =. 2025 , eprint =
2025
-
[12]
Nature Physics , volume =
Wills, Adam and Hsieh, Min-Hsiu and Yamasaki, Hayata , title =. Nature Physics , volume =. 2025 , doi =
2025
-
[13]
Proceedings of the 57th Annual ACM Symposium on Theory of Computing , pages =
Golowich, Louis and Guruswami, Venkatesan , title =. Proceedings of the 57th Annual ACM Symposium on Theory of Computing , pages =. 2025 , isbn =. doi:10.1145/3717823.3718234 , abstract =
-
[14]
Efficiently List-Decodable Punctured Reed-Muller Codes , year=
Guruswami, Venkatesan and Jin, Lingfei and Xing, Chaoping , journal=. Efficiently List-Decodable Punctured Reed-Muller Codes , year=
-
[15]
Physical Review Letters , volume =
Anirudh Krishna and Jean-Pierre Tillich , title =. Physical Review Letters , volume =. 2019 , month =
2019
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.